A ray of light moving parallel to x-axis gets reflected from a parabolic mirror `(y-2)^(2)=4(x+1)`. After reflection the ray must pass through

Options

A. `(0,2)`
B. (0,-2)
C. (2,0)
D. (1,2)

Answer

To solve the problem step by step, we need to find the focus of the given parabolic mirror and determine the point through which the reflected ray passes.

### Step 1: Write down the equation of the parabola
The given equation of the parabolic mirror is:
\[ (y - 2)^2 = 4(x + 1) \]

### Step 2: Identify the standard form of the parabola
The standard form of a parabola that opens to the right is:
\[ Y^2 = 4aX \]
where \( (0, 0) \) is the vertex and \( (a, 0) \) is the focus.

### Step 3: Rewrite the given equation in standard form
We can rewrite the given equation to match the standard form:
\[ (y - 2)^2 = 4(x + 1) \]
Let \( Y = y - 2 \) and \( X = x + 1 \). Then, the equation becomes:
\[ Y^2 = 4X \]
Here, \( 4a = 4 \) implies \( a = 1 \).

### Step 4: Determine the vertex and focus
From the standard form \( Y^2 = 4aX \), we know:
- The vertex of the parabola is at \( (0, 0) \).
- The focus is at \( (a, 0) \), which translates to \( (1, 0) \) in the transformed coordinates.

### Step 5: Convert the focus back to original coordinates
Since we made substitutions, we need to convert the focus back to the original coordinates:
- The focus in terms of \( x \) and \( y \) is:
\[ X = 1 \implies x + 1 = 1 \implies x = 0 \]
\[ Y = 0 \implies y - 2 = 0 \implies y = 2 \]
Thus, the focus in the original coordinates is:
\[ (0, 2) \]

### Step 6: Conclusion
The reflected ray from the parabolic mirror will pass through the focus. Therefore, the ray must pass through the point:
\[ \boxed{(0, 2)} \]